The WS transform for the Kuramoto model with distributed amplitudes, phase lag and time delay

Abstract

We apply the Watanabe–Strogatz (WS) transform to a generalized Kuramoto model with distributed parameters describing the amplitude of oscillation, phase lag, and time delay at each node of the system. The model has global coupling and identical frequencies, but allows for repulsive interactions at arbitrary nodes leading to conformist-contrarian phenomena together with variable amplitude and time-delay effects. We show how to determine the initial values of the WS system for any initial conditions for the Kuramoto system, and investigate the asymptotic behaviour of the WS variables. For the case of zero time delay the possible asymptotic configurations are determined by the sign of a single parameter μ which measures whether or not the attractive nodes dominate the repulsive nodes. If the system completely synchronizes from general initial conditions, whereas if one of two types of phase-locked synchronization occurs, depending on the initial values, while for periodic solutions can occur. For the case of arbitrary non-uniform time delays we derive a stability condition for completely synchronized solutions.